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Beliebt Rechnen >

integral of 10x(ln(x))^2

Lösung

\int 10 x (ln (x))^{2} d x

Lösung

10 (\frac{1}{2} x^{2} ln^{2} (x) - \frac{1}{2} x^{2} ln (x) + \frac{x ^{2}}{4}) + C

Schritte zur Lösung

\int 10 x (ln (x))^{2} d x

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 10 \cdot \int x (ln (x))^{2} d x

Wende die partielle Integration an

= 10 (\frac{1}{2} x^{2} ln^{2} (x) - \int x ln (x) d x)

\int x ln (x) d x = \frac{1}{2} x^{2} ln (x) - \frac{x ^{2}}{4}

= 10 (\frac{1}{2} x^{2} ln^{2} (x) - (\frac{1}{2} x^{2} ln (x) - \frac{x ^{2}}{4}))

Vereinfache

= 10 (\frac{1}{2} x^{2} ln^{2} (x) - \frac{1}{2} x^{2} ln (x) + \frac{x ^{2}}{4})

Füge eine Konstante zur Lösung hinzu

= 10 (\frac{1}{2} x^{2} ln^{2} (x) - \frac{1}{2} x^{2} ln (x) + \frac{x ^{2}}{4}) + C

Graph

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